Nonlinear maps preserving the mixed product $[A\bullet B, C]_{*}$ on von Neumann algebras

Abstract

Let $\mathcal {A}$ and $\mathcal {B}$ be two von Neumann
algebras. For $A, B\in\mathcal {A}$, define by $[A, B]_{*}=AB-BA^{\ast}$ and $A\bullet B=AB+BA^{\ast}$ the new products of $A$ and $B$. Suppose that a bijective map $\Phi: \mathcal {A}\rightarrow \mathcal {B}$ satisfies$\Phi([A\bullet B,C]_{*})=[\Phi(A)\bullet\Phi(B),\Phi(C)]_{*}$ for all $A, B,C\in\mathcal {A}$. In this paper, it is proved that if $\mathcal {A}$ and $\mathcal {B}$ be two von Neumann algebras with no central abelian projections, then the map $\Phi(I)\Phi$ is a sum of a linear $*$-isomorphism and a conjugate linear $*$-isomorphism,
where $\Phi(I)$ is a self-adjoint central element in $\mathcal {B}$ with $\Phi(I)^{2}=I.$ If $\mathcal {A}$ and $\mathcal {B}$ are two factor von Neumann algebras, then $\Phi$ is a linear $\ast$-isomorphism, or a conjugate linear $\ast$-isomorphism, or the
negative of a linear $\ast$-isomorphism, or the negative of a conjugate linear $\ast$-isomorphism.